Theorem rexv 2821

Description: An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
rexv	|- ( E. x e. _V ph <-> E. x ph )

Proof of Theorem rexv

Step	Hyp	Ref	Expression
1		df-rex 2516	. 2 |- ( E. x e. _V ph <-> E. x ( x e. _V /\ ph ) )
2		vex 2805	. . . 4 |- x e. _V
3	2	biantrur 303	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	exbii 1653	. 2 |- ( E. x ph <-> E. x ( x e. _V /\ ph ) )
5	1, 4	bitr4i 187	1 |- ( E. x e. _V ph <-> E. x ph )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1540   e. wcel 2202   E.wrex 2511   _Vcvv 2802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-rex 2516  df-v 2804

This theorem is referenced by:  rexcom4  2826  spesbc  3118  abnex  4544  dfco2  5236  dfco2a  5237  finacn  7418